Level Raising via Unitary Shimura Varieties with Good Reduction and an Ihara Lemma

Recall a classical theorem of Ribet: Fix a prime l; consider a weight-2 level-N newform f satisfying the mod l level-raising condition at a prime p coprime to Nl. Then Ribet shows that the first Galois cohomology of the mod l Galois representation of Q_p associated with f can be realized as the Abel-Jacobi image of the supersingular locus of the level-N modular curve over F_p. In this talk, we will discuss how one can generalize this phenomenon to higher-dimensional unitary Shimura varieties at inert places (which remains a conjecture in general), and its relation with a certain Ihara type lemma for such varieties. We will explain cases for which we have confirmed such conjecture; and if time permits, we will mention its number-theoretical implications. This is a joint work in progress with Yichao Tian and Liang Xiao.

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